The Number Line & Ordering Integers
The number line is the simplest picture in all of math: numbers laid out on a straight line, increasing from left to right. Zero sits in the middle, positive numbers stretch to the right, and negative numbers stretch to the left. Every comparison question — which number is bigger, which is smaller — is answered by that picture.
Ordering integers trips people up in exactly one place: negatives. It feels like should be bigger than because is bigger than . The number line settles it — sits farther left, so is smaller. Learn to trust position instead of instinct and this skill is automatic.
Left means smaller
On a number line, the farther left a number sits, the smaller it is. The farther right, the bigger. That one sentence is the whole rule, and it works for every pair of numbers — positive, negative, or zero.
Look at and below. is far to the left of , so . You never have to think about signs separately; position handles it.
Comparing negatives
With two negative numbers, the one closer to zero is greater. is closer to zero than , so . A useful real-world anchor: owing 1 dollar leaves you better off than owing 8 dollars, and a temperature of degrees is warmer than degrees.
The inequality symbols read like an arrow that opens toward the bigger number: says the small end points at , the open end faces . If you can place both numbers on the line, the symbol writes itself.
Ordering a whole list
To order a list of integers from least to greatest, picture (or sketch) each one on the number line, then read them off from left to right. Negatives come first, with the most negative — the one farthest from zero — leading the list. Then zero, then the positives from smallest to largest.
A quick sanity check when you finish: the most negative number should be first and the largest positive number should be last. If either end looks wrong, one number got placed on the wrong side of zero.
Worked examples
Example 1: comparing a negative and a positive
Compare and using or .
Answer:
Example 2: comparing two negatives
Compare and using or .
Answer:
Example 3: ordering a list
Order from least to greatest.
Answer:
Try one yourself
Common questions
Is smaller than ?
Yes. sits farther left on the number line, so it is smaller: . With negatives, a bigger-looking digit means a smaller number, because it is farther from zero in the negative direction.
Which way does the inequality symbol point?
The open side faces the bigger number and the point aims at the smaller one. In , the point aims at (smaller) and the symbol opens toward (bigger).
Is zero positive or negative?
Neither. Zero is the dividing point between the negatives and the positives. When you order a list, zero goes after every negative number and before every positive number.
What is the fastest way to order a mixed list?
Split the list into negatives and positives. Order the negatives by putting the one farthest from zero first, then write zero if it appears, then order the positives smallest to largest.
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